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52     ON THE EFFECT OF THE INTERNAL  FRICTION  OF FLUIDS
M\, Me) and 1 - M'e by *r/4, and by L, as well as in reducing the right-hand member of (105) to the form k + J~^l ti. The terms of the series (113) were calculated to five places of decimals. That these series are sufficiently convergent to be employed when m-1'5, might be presumed from the numerical values of the terms, and is confirmed by finding that they give k = 1*952, and Jc' ==1*153. For m = r5 and a few of the succeeding values, the second and third of the series (113) were summed directly as far as m"5 inclusively, and the remainders were calculated from the formulae (114). Two columns are annexed, which give the values of m% and ttl2&', and exhibit the law of the variation of the two parts of the force F, when the radius of the cylinder varies, the nature of the fluid and time of oscillation remaining unchanged. Four significant figures are retained in all the results.
m	k	kf	m2/c	m2#	m	Jc	W	m2&	m2/c'
0	00	oo	0	0	2-1	1-677	7822	7*395	3-450
i	19-70	48-63	1970	4863	2-2	1-646	7421	7-966	3-592
2	9-166	16-73	3666	6691	2-3	1-618	7059	8-557	3-734
3	6-166	9-258	5549	8332	2-4	1-592	6730	9-168	3-877
4	4-771	6-185	7633	9896	2-5	1-568	6430	9-799	4-019
5	3-968	4-567	9920	1-142	2-6	1-546	6154	10-45	4160
6	3-445	3-589	1-240	1-292	2-7	1-526	5902	11-12	4-303
7	3-082	2-936	1-510	1-439	2-8	1-507	5669	11-81	4-444
8	2-812	2-477	1-800	1-585	2-9	1-489	5453	12-52	4-586
9	2-604	2-137	2-110	1-731	3-0	1-473	5253	13-25	4-728
1-0	2-439	1-876	2-439	1-876	3-1	1-457	5068	14-01	4-870
1-1	2-306	1-678	2-790	2-021	3-2	1-443	4895	14-78	5-012
1-2	2-194	1-503	3-160	2-164	3-3	1-430	4732	15-57	5-154
1-3	2-102	1-365	3-552	2-307	3-4	1-417	4581	16-38	5-296
1-4	2-021	1-250	3-961	2-450	3-5	1-405	4439	17-21	5-437
1-5	1-951	1-153	4-389	2-595	3-6	1-394	4305	18-06	5-580
1-6	1-891	1-069	4-841	2-739	3-7	1-383	4179	18-93	5-721
1-7	1-838	9965	5-312	2-880	3-8	1-373	4060	19-82	5-863
1-8	1-791	9332	5-804	3-024	3-9	1-363	3948	20-73	6-005
1-9	1-749	8767	6-314	3-165	4-0	1-354	3841	21-67	6-145
2'0	1-711	8268	6-845	3-307	00	1	0	00	oo
The numerical calculation by means of the formulas (103), (104), (105) becomes very laborious when many values of the functions are required. The difficulty of the calculation increases with the value of m for two reasons, first, the calculation of the functions M"0, &c. becomes longer, and secondly, the moduli of the numerator and denominator of the fraction in the right-hand